3. The case of the Crab Nebula
For the Crab Nebula pulsar we have in Eq. (1) , , where the braking index n is (Groth 1975). For the parameters of the injected spectrum (Eq. (13)), following a procedure similar to PS, we derive from the observations the values , , . The parameter in Eq. (14) determines, through Eq. (5), the present value of B; we take the latter to be equal to , so that . Given these parameters the only unknowns in Eq. (16) are the injection radius and , the latter containing the cut-off energies of the injected spectrum. These parameters have very different observational signatures: the first one is related to the radial distance from the central pulsar of the luminosity peak, and the second one only affects the overall nebular synchrotron flux.
The model predictions for the synchrotron surface brightness profile of the Crab Nebula have been compared with high resolution data at various frequencies.
For the radio band we have used a VLA map at a frequency of 1.4 GHz (Bietenholz et al. 1997).
The spatial and spectral distribution of the optical synchrotron continuum was determined by Véron-Cetty & Woltjer (1993) after subtraction of the thermal contributions from foreground stars and filaments, from four narrow-band images at wavelenghts of 9241, 6450, 5364 and 3808 Å. We have reanalysed these maps, kindly put at our disposal by M.P. Véron-Cetty, with state-of-the-art star subtraction algorithms.
Finally, in the X-ray band, we have used, after deconvolution of the instrumental PSF and subtraction of the dust halo (Bandiera et al. 1998), a collection of all the public ROSAT HRI data concerning the Crab Nebula. We have estimated the mean photon energy of these data to be 1 keV.
In our model the synchrotron surface brightness is simply obtained by integration of Eq. (2) along the line of sight:
with the expression for the particle number density N in calculated from Eq. (16).
In order to compare our spherical model with the observations, we have extracted from each image what we call a "radial intensity profile": we first sampled the emission profiles of the nebula along different directions, taking the mean values over small areas of and then, after rescaling the different axes to a common length, we averaged those profiles. The procedure just described, to which we refer in the following as "data sphericization", is the main cause of uncertainty in our radial profiles and it is what we take into account in the error bars attached to the data points in the following plots.
When the radiative losses are negligible ( in Eq. (16) throughout the entire nebula) and B enter the expression of the synchrotron emissivity simply as multiplying factors. Therefore fitting the shape of the surface brightness profile at radio frequencies allows a straightforward determination of , and once this is known, the value of simply comes from fitting the integrated flux, so that the model is fully determined.
Our best fit estimate gave , which, for a 2 kpc distance to the Crab Nebula, translates into pc. The corresponding radial profile at 1.4 GHz is plotted against the data in Fig. 1. This value of yields a distance from the pulsar to the injection site fully compatible with the association between the particle acceleration region and the location of the optical wisps.
As in the PS model, the integrated fluxes reproduce the observations at all frequencies: both the solid and the dashed curve yield the same flux as the interpolation of the data, when integrated on a spherical surface of radius 2 pc. Nevertheless, although in the radio part of the spectrum the fit to the observed profile is rather good and substantially improves the homogeneous model, the model predictions fail to reproduce the data at optical and X-ray wavelengths.
As shown in Fig. 2 the emission at optical and X-ray wavelengths calculated on the basis of the present model is too concentrated: the highest energy particles emit most of their energy immediately after the injection. This causes most of the flux to originate from a narrow region and the particles to travel a very short distance from the injection site before their energy is degraded by severe synchrotron losses.
This effect could be cured by substantially lowering the magnetic field and thereby the synchrotron losses. However the inverse Compton data do not allow this. Moreover if the magnetic field energy is less than the particle energy the model becomes invalid and at lower fields the total particle energy would soon exceed that produced by the pulsar.
The homogeneous PS model in which the particles move freely through the Nebula yields too broad a distribution of the emissivity, our model a too narrow one at the higher frequencies. Apparently some of the particles can reach the outer parts of the nebula without suffering the large synchrotron losses which occur when they are fully tied to the field lines.
Diffusion of particles could solve this problem, but the diffusion coefficient would have to be times larger than for Bohm diffusion, in agreement with previous estimates (Wilson 1972b). A more complex field structure in which some lines connect the inner and outer parts of the nebula might give an acceptable solution, since the particles could move along these lines at a substantial fraction of the speed of light.
© European Southern Observatory (ESO) 2000
Online publication: July 13, 2000